Since \({\rm Gr}_{k,n}^{>0}\)  and \({\rm Mat}_{n,k+m}^{>0}\)  are path connected (see [1]} for the positive Grassmannian), there exists a continuous
path
\(\left(C,\mathcal {Z}\right):\left[0,1\right]\rightarrow {\rm Gr}_{k,n}^{>0}\times {\rm Mat}_{n,k+m}^{>0}\) 
